SYDNEY (Reuters) - Australian police on Sunday (July 3) said they are considering an investigation into allegations of political skullduggery involving text messages sent to voters as the nation went to the polls on the weekend.

The messages, which appeared on voters' phones as though they had come from state health insurer Medicare, warned that public health services would be privatised under a Liberal Party-led coalition government, although the conservative ruling party says it has no such policy.

Prime Minister Malcolm Turnbull called the message "an extraordinary act of dishonesty" in a speech to party faithful on election night, as counting from Saturday's poll showed the government was at risk of losing its majority.

"It's a pretty shameful episode in Australian political history," he said, laying the blame on the opposition Labor Party.

The Queensland state branch of Labor told Fairfax Media on Saturday that it had sent the messages, and said they was not meant to appear as though they had been sent by Medicare.

The text message said: "Time is running out to save Medicare," according to screenshots published by Fairfax and News Corp.

Labor made the prospect of privatisation of Medicare a key point of attack in its election campaign.

The Australian Federal Police confirmed that the matter had been referred to them.

"This matter is now being evaluated and whilst this occurs it would not be appropriate to provide further comment," the force said in a statement on Sunday.